Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1045st+ 1 −st=λ 1 +θ 1 (ft−st)+[λ 2 +θ 2 (ft−st)]
×!(
st−ft− 1 ;γ)- (^) t+ 1 , (22.48)
where!
(
st−ft− 1 ;γ
)
= 1 −exp
(
−γ(st−ft− 1 )^2
)
. Equivalently:
st+ 1 −ft=λ 1 +(θ 1 − 1 )(ft−st)+[λ 2 +θ 2 (ft−st)]!(
(st−ft− 1 );γ)- (^) t+ 1. (22.49)
Whenst−ft− 1 is small we obtain from (22.48):
st+ 1 −st=λ 1 +θ 1 (ft−st), (22.50)
and whenst−ft− 1 is large:
st+ 1 −st=λ 1 +λ 2 +(θ 1 +θ 2 )(ft−st). (22.51)
Sarnoet al.(2006) report that the restrictionθ 1 +θ 2 =1 cannot be rejected for
the currencies they examine and also thatθ 1 <0. Simulated data from this model
generate negative estimates ofθin the standard regressions (22.36) and (22.37). In
fact, if the constraint thatθ 1 +θ 2 =1 is imposed, the model collapses to the form
employing expected excess returns as:
st+ 1 −ft=−θ 2 (ft−st)exp
(
−γ
[
Et(st+ 1 −ft)
] 2 ) - (^) t+ 1 , (22.52)
whenλ 1 +λ 2 =0, whereEt(st+ 1 −ft)is the expected excess return formed on
information available in periodt. In this form the model allows expectations to be
formed rationally, as in the arbitrage consistent STAR (ARBSTAR) model proposed
by Peel and Venetis (2005), which remedies some economic difficulties when TAR,
ESTAR, or LSTAR models are employed to model arbitrage.^47
The estimates of Sarnoet al.(2006)^48 suggest that the relationship between the
excess return and the forward premium is nonlinear. However, it is not clear that
the efficiency proposition is tested, or indeed that there is a well defined null, except
under the assumption of risk neutrality. Time-varying risk premia will imply the
lack of a unit relationship in the outer regimes. It would be of interest to estimate
the Sarnoet al.(2006) model when risk premia area priorismall – e.g., in credible
periods in the Exchange Rate Mechanism (ERM), as in the analysis of Flood and
Rose (1996) mentioned above.
An extension of the nonlinear STAR model has recently been employed to model
forward premia and PPP deviations (see Smallwood, 2005; Baillie and Kapetanios,
2005, 2006).^49 The idea is that the fractional difference of a series is an ESTAR or
LSTAR model, and the model is called FI-STAR. In particular, the FI-STAR model for
a time seriesytis defined as:
( 1 −L)dyt=
⎛
⎝α1,0+
∑p
j= 1
α1,j( 1 −L)dyt−j
⎞
⎠
⎛
⎝α2,0+
∑p
j= 1
α2,j( 1 −L)dyt−j
⎞
⎠G(yt−k;γ,c)+εt, (22.53)